On the energy of inviscid singular flows

AbstractIt is known that the energy of a weak solution to the Euler equation is conserved if it is slightly more regular than the Besov space B3,∞1/3. When the singular set of the solution is (or belongs to) a smooth manifold, we derive various Lp-space regularity criteria dimensionally equivalent to the critical one. In particular, if the singular set is a hypersurface the energy of u is conserved provided the one-sided non-tangential limits to the surface exist and the non-tangential maximal function is L3 integrable, while the maximal function of the pressure is L3/2 integrable. The results directly apply to prove energy conservation of the classical vortex sheets in both 2D and 3D at least in those cases where the energy is finite